Weak stability of nonuniformly stable multidimensional shocks

SIAM Journal on Mathematical Analysis

Volumn 34, Issue 1, 2003, Pages 142-172

  Coulombel, Jean François ^a  

^a UNIVERSITÉ DE LYON   (France)

Author keywords

Conservation laws; Multidimensional shocks; Phase transitions; Stability of shock waves

Indexed keywords

EID: 0037287376     PISSN: 00361410     EISSN: None     Source Type: Journal    
DOI: 10.1137/S0036141001392803     Document Type: Article

References (40)

1. M. S. AGRANOVIČ, Boundary value problems for systems with a parameter, Math. USSR-Sb., 13 (1971), pp. 25-64.
2. V. I. ARNOL'D, Ordinary Differential Equations, Springer-Verlag, Berlin, 1992.
3. H. BEIRÃO DA VEIGA, On the existence theorem for the barotropic motion of a compressible inviscid fluid in the half-space, Ann. Mat. Pura Appl. (4), 163 (1993), pp. 265-289.
4. S. BENZONI-GAVAGE, Stability of multi-dimensional phase transitions in a van der Waals fluid, Nonlinear Anal., 31 (1998), pp. 243-263.
5. S. BENZONI-GAVAGE, Stability of subsonic planar phase boundaries in a van der Waals fluid, Arch. Ration. Mech. Anal., 150 (1999), pp. 23-55.
6. A. M. BLOKHIN, Estimation of the energy integral of a mixed problem for gas dynamics equations with boundary conditions on the shock wave, Sibirsk. Mat. Zh., 22 (1981), pp. 23-51, 229 (in Russian).
7. A. M. BLOKHIN, Uniqueness of the classical solution of a mixed problem for equations of gas dynamics with boundary conditions on a shock wave, Sibirsk. Mat. Zh., 23 (1982), pp. 17-30, 222 (in Russian).
8. J. CHAZARAIN AND A. PIRIOU, Introduction to the Theory of Linear Partial Differential Equations, North-Holland, Amsterdam, 1982.
9. R. COURANT AND D. HILBERT, Methods of Mathematical Physics, Vol. 2: Partial Differential Equations, Interscience, New York, London, 1962.
10. R. COURANT AND K. O. FRIEDRICHS, Supersonic Flow and Shock Waves, Interscience, New York, 1948.
11. W. DOMAŃSKI, Surface and boundary waves for linear hyperbolic systems: Applications to basic equations of electrodynamics and mechanics of continuum, J. Tech. Phys., 30 (1989), pp. 283-300.
12. S. P. D'YAKOV, On the stability of shock waves, Ž. Èksper. Teoret. Fiz., 27 (1954), pp. 288-295 (in Russian).
13. J. ERPENBECK, Stability of step shocks, Phys. Fluids, 5 (1962), pp. 1181-1187.
14. H. FREISTÜHLER, The persistence of ideal shock waves, Appl. Math. Lett., 7 (1994), pp. 7-11.
15. O. GUÈS, Problème mixte hyperbolique quasi-linéaire caractéristique, Comm. Partial Differential Equations, 15 (1990), pp. 595-645.
16. R. HERSH, Mixed problems in several variables, J. Math. Mech., 12 (1963), pp. 317-334.
17. R. L. HIGDON, Initial-boundary value problems for linear hyperbolic systems, SIAM Rev., 28 (1986), pp. 177-217.
18. D.-Y. HSIEH AND X.-P. WANG, Phase transition in van der Waals fluid, SlAM J. Appl. Math., 57 (1997), pp. 871-892.
19. V. M. KONTOROVIČ, Stability of shock waves in relativistic hydrodynamics, Soviet Physics JETP, 34 (1958), pp. 127-132.
20. H. O. KREISS, Initial boundary value problems for hyperbolic systems, Comm. Pure Appl. Math., 23 (1970), pp. 277-298.
21. P. D. LAX, Hyperbolic systems of conservation laws. II, Comm. Pure Appl. Math., 10 (1957), pp. 537-566.
22. P. D. LAX, Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves, CBMS-NSF Regional Conf. Ser. in Appl. Math. 11, SIAM, Philadelphia, 1973.
23. T. T. LI AND W. C. YU, Boundary Value Problems for Quasilinear Hyperbolic Systems, Duke University Mathematics Department, Durham, NC, 1985.
24. A. MAJDA, The existence of multi-dimensional shock fronts, Mem. Amer. Math. Soc., 43 (1983).
25. A. MAJDA, The stability of multi-dimensional shock fronts, Mem. Amer. Math. Soc., 41 (1983).
26. A. MAJDA, Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables, Springer-Verlag, New York, 1984.
27. G. MÉTIVIER, Problèmes mixtes non linéaires et stabilité des chocs multidimensionnels, Astérisque, 152-153 (1987), pp. 37-53.
28. G. MÉTIVIER, The block structure condition for symmetric hyperbolic systems, Bull. London Math. Soc., 32 (2000), pp. 689-702.
29. S. MIYATAKE, Neumann Operator for Wave Equation in a Half Space and Microlocal Orders of Singularities Along the Boundary, Séminaire sur les Équations aux Dérivées Partielles, 1992-1993, Exp. No. XIV, École Polytechnique, Palaiseau, Cedex, France, 1993.
30. A. MOKRANE, Problèmes mixtes hyperboliques non-linéaires, Ph.D. Thesis, Université de Rennes I, Rennes Cedex, France, 1987.
31. 2-well-posed mixed problems for hyperbolic systems of first order, Hokkaido Math. J., 4 (1975), pp. 82-158.
32. J. V. RALSTON, Note on a paper of Kreiss, Comm. Pure Appl. Math., 24 (1971), pp. 759-762.
33. R. D. RICHTMYER, Taylor instability in shock acceleration of compressible fluids, Comm. Pure Appl. Math., 13 (1960), pp. 297-319.
34. M. SABLÉ-TOUGERON, Existence pour un problème de l'élastodynamique Neumann non linéaire en dimension 2, Arch. Ration. Mech. Anal., 101 (1988), pp. 261-292.
35. P. SECCHI, Inflow-outflow problems for inviscid compressible fluids, Commun. Appl. Anal., 2 (1998), pp. 81-110.
36. D. SERRE, Systems of Conservation Laws. 1, Cambridge University Press, Cambridge, UK, 1999.
37. D. SERRE, Systems of Conservation Laws. 2, Cambridge University Press, Cambridge, UK, 2000.
38. M. SHEARER, Admissibility criteria for shock wave solutions of a system of conservation laws of mixed type, Proc. Roy. Soc. Edinburgh Sect. A, 93 (1982/83), pp. 233-244.
39. M. SLEMROD, Admissibility criteria for propagating phase boundaries in a van der Waals fluid, Arch. Ration. Mech. Anal., 81 (1983), pp. 301-315.
40. L. TRUSKINOVSKY, About the "normal growth" approximation in the dynamical theory of phase transitions, Contin. Mech. Thermodyn., 6 (1994), pp. 185-208.